Семин В.В., Горохов А.В. Немарковская релаксация квантовых

¨¸Ó³ ¢ —Ÿ. 2011. ’. 8, º 5(168). ‘. 785Ä789
ŠŒœ’…›… ’…•‹ƒˆˆ ‚ ”ˆ‡ˆŠ…
…ŒŠ‚‘ŠŸ …‹Š‘–ˆŸ Š‚’‚›•
‘ˆ‘’…Œ ˆ ‚›—ˆ‘‹…ˆ… ”Œ›
‘…Š’‹œ›• ‹ˆˆ‰
‚. ‚. ‘¥³¨´, . ‚. ƒμ·μÌμ¢
‘ ³ ·¸±¨° £μ¸Ê¤ ·¸É¢¥´´Ò° Ê´¨¢¥·¸¨É¥É, ‘ ³ · , μ¸¸¨Ö
‚ ¸É ÉÓ¥ ¢Ò¢μ¤¨É¸Ö ´¥±μÉμ·μ¥ ´¥³ ·±μ¢¸±μ¥ ±¨´¥É¨Î¥¸±μ¥ Ê· ¢´¥´¨¥ ¤²Ö ¸¨¸É¥³Ò ¤¢ÊÌ ¨¤¥´É¨Î´ÒÌ ¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì ¤¢ÊÌÊ·μ¢´¥¢ÒÌ Éμ³μ¢, ´ μ¸´μ¢ ´¨¨ ·¥Ï¥´¨Ö ±μÉμ·μ£μ ¢ÒΨ¸²Ö¥É¸Ö
Ëμ·³ ¸¶¥±É· ²Ó´ÒÌ ²¨´¨° ¤²Ö ¤ ´´μ° ¸¨¸É¥³Ò.
A non-Markovian kinetic equation for a system of two identical interacting two-level atoms has
been derived. Solution of this equation has been used for calculation of spectral lines shape of this
system.
PACS: 32.30.-r
‚‚…„…ˆ…
ˆ§ÊÎ¥´¨¥ ¸¶¥±É· ²Ó´ÒÌ ¸¢μ°¸É¢ ¨§²ÊÎ¥´¨Ö Éμ³μ¢ ¢μ ¢´¥Ï´¨Ì ¶μ²ÖÌ ´¥¸¥É ¢ ¦´ÊÕ
¨´Ëμ·³ Í¨Õ μ ¸É·Ê±ÉÊ·¥ Ô´¥·£¥É¨Î¥¸±¨Ì Ê·μ¢´¥° ¨ ¤¨´ ³¨±¥ ¶¥·¥Ìμ¤μ¢ ³¥¦¤Ê ´¨³¨.
‘ ¤·Ê£μ° ¸Éμ·μ´Ò, ¤¢ÊÌÊ·μ¢´¥¢Ò¥ Éμ³Ò ³μ£ÊÉ · ¸¸³ É·¨¢ ÉÓ¸Ö ± ± ±Ê¡¨ÉÒ Å ´μ¸¨É¥²¨ ¨´Ëμ·³ ͨ¨ ¢ ±¢ ´Éμ¢μ³ ±μ³¶ÓÕÉ¥·¥. Éμ ¤¥² ¥É ´¥μ¡Ì줨³Ò³ ¤¥É ²Ó´μ¥ ¨§ÊÎ¥´¨¥
¸¶¥±É· ²Ó´ÒÌ Ì · ±É¥·¨¸É¨± ¨§²ÊÎ¥´¨Ö ¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì Éμ³μ¢ ¢ · §²¨Î´ÒÌ ¶μ²ÖÌ.
’· ¤¨Í¨μ´´μ ¤²Ö ·¥Ï¥´¨Ö ¶μ¤μ¡´ÒÌ § ¤ Î ¨¸¶μ²Ó§ÊÕÉ ¶μ¤Ìμ¤ ±¢ ´Éμ¢μ° É¥μ·¨¨ μɱ·ÒÉÒÌ ¸¨¸É¥³ [1], ¶·¨ ÔÉμ³ ±²ÕÎ¥¢Ò³ ¶·¨ ¢Ò¢μ¤¥ ±¨´¥É¨Î¥¸±¨Ì Ê· ¢´¥´¨° Ö¢²Ö¥É¸Ö
¶·¥¤¶μ²μ¦¥´¨¥ μ ³ ·±μ¢μ¸É¨ ¶·μÍ¥¸¸ ¢§ ¨³μ¤¥°¸É¢¨Ö Éμ³μ¢ ¸μ ¸¢μ¨³ μ±·Ê¦¥´¨¥³,
É. ¥. ¶·¥´¥¡·¥¦¥´¨¥ ÔËË¥±É ³¨ ¶ ³Öɨ. ˆ§¢¥¸É´Ò¥ ¨§ ²¨É¥· ÉÊ·Ò ´¥³ ·±μ¢¸±¨¥ ±¨´¥É¨Î¥¸±¨¥ Ê· ¢´¥´¨Ö ´¥ ²¨Ï¥´Ò ´¥¤μ¸É ɱμ¢. ¨¡μ²¥¥ μ¡Ð¥¥ Ê· ¢´¥´¨¥ ± Ϩ³ Ä
–¢ ´Í¨£ [2] ¶·¥¤¸É ¢²Ö¥É ¸±μ·¥¥ Ëμ·³ ²Ó´Ò° ¨´É¥·¥¸, ¶μ¸±μ²Ó±Ê ¥£μ ´¥¢μ§³μ¦´μ · §·¥Ï¨ÉÓ. „·Ê£¨¥ Ê· ¢´¥´¨Ö Ö¢²ÖÕÉ¸Ö ¸¶· ¢¥¤²¨¢Ò³¨ ²¨ÏÓ ¤²Ö μÎ¥´Ó ʧ±μ£μ ±² ¸¸ ¶·μÍ¥¸¸μ¢ [3] ²¨¡μ Ö¢²ÖÕÉ¸Ö Ë¥´μ³¥´μ²μ£¨Î¥¸±¨³¨ [4], ÔËË¥±ÉÒ ¶ ³Öɨ ¢ ±μÉμ·ÒÌ μ¶¨¸Ò¢ ÕÉ¸Ö ¢¢¥¤¥´¨¥³ ¤μ¶μ²´¨É¥²Ó´ÒÌ ³´μ¦¨É¥²¥° ¢ Ê· ¢´¥´¨¨ ‹¨´¤¡² ¤ [5].
–¥²Ó ´ ¸ÉμÖÐ¥° ¸É ÉÓ¨ Å ¢Ò¢μ¤ ´¥±μÉμ·μ£μ μ¡μ¡Ð¥´¨Ö Ê· ¢´¥´¨Ö ‹¨´¤¡² ¤ ¤²Ö
¸¨¸É¥³Ò ¤¢ÊÌ ¤¨¶μ²Ó-¤¨¶μ²Ó´μ-¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì ¤¢ÊÌÊ·μ¢´¥¢ÒÌ Éμ³μ¢ ¸ ÊÎ¥Éμ³ ¶·μÍ¥¸¸μ¢ ¢§ ¨³μ¤¥°¸É¢¨Ö ¸ É¥¶²μ¢Ò³ ·¥§¥·¢Ê ·μ³ ¸ ±· É±μ¢·¥³¥´´μ° ¶ ³ÖÉÓÕ. μ¸´μ¢¥
·¥Ï¥´¨Ö ¶μ²ÊÎ¥´´μ£μ Ê· ¢´¥´¨Ö ¸É·μ¨É¸Ö ±μ´ÉÊ· ²¨´¨¨ ¨§²ÊÎ¥´¨Ö ¤ ´´μ° ¸¨¸É¥³Ò ¨
¨¸¸²¥¤Ê¥É¸Ö ¥£μ μɲ¨Î¨¥ μÉ ±μ´ÉÊ· ²¨´¨¨, · ¸¸Î¨É ´´μ£μ ¢ ³ ·±μ¢¸±μ³ ¶·¨¡²¨¦¥´¨¨.
786 ‘¥³¨´ ‚. ‚., ƒμ·μÌμ¢ . ‚.
1. Œ„…‹œ ˆ Šˆ…’ˆ—…‘Š… “‚…ˆ…
¸¸³μÉ·¨³ ¤¢ ¤¨¶μ²Ó-¤¨¶μ²Ó´μ-¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì ¤¢ÊÌÊ·μ¢´¥¢ÒÌ Éμ³ ¢ É¥¶²μ¢μ³ ·¥§¥·¢Ê ·¥. ƒ ³¨²ÓÉμ´¨ ´ É ±μ° ¸¨¸É¥³Ò
H = HA + HT + Hint + HAA + HAF ,
HA = ω0
(1)
σpz Å £ ³¨²ÓÉμ´¨ ´ ¸¢μ¡μ¤´ÒÌ Éμ³μ¢, ω0 Å Î ¸ÉμÉ ¶¥·¥Ìμ¤μ¢ ¢ Éμ³¥,
σpz Å ¤¨ £μ´ ²Ó´Ò° £¥´¥· Éμ· £·Ê¶¶Ò SU (2); HT = ωk b+
k bk Å £ ³¨²ÓÉμ´¨ ´ É¥·³μp
k
¸É É (É¥¶²μ¢μ£μ ·¥§¥·¢Ê · ), ωk Å Î ¸ÉμÉ b+
k ¨ bk Å μ¶¥· Éμ·Ò ·μ¦¤¥´¨Ö
k-£μ ËμÉμ´ ,
+ ikRp
+ h. c.) Å £ ³¨²ÓÉμ´¨ ´ ¢§ ¨¨ Ê´¨ÎÉ즥´¨Ö k-£μ ËμÉμ´ ; Hint = (gkp bk σp e
k,p
³μ¤¥°¸É¢¨Ö Éμ³ ¨ É¥·³μ¸É É , £¤¥ gkp Å ¸μμÉ¢¥É¸É¢ÊÕШ¥ ±μ´¸É ´ÉÒ ¢§ ¨³μ¤¥°¸É¢¨Ö,
σp± Å ¶μ¢ÒÏ ÕШ° ¨ ¶μ´¨¦ ÕШ° Éμ³´Ò¥ 춥· Éμ·Ò, Rp Å · ¤¨Ê¸-¢¥±Éμ· k-£μ Éμ³ ,
k Å ¢μ²´μ¢μ° ¢¥±Éμ·; HAA =
Vpp σp+ σp− Å £ ³¨²ÓÉμ´¨ ´ ¤¨¶μ²Ó-¤¨¶μ²Ó´μ£μ ¢§ ¨p=p
³μ¤¥°¸É¢¨Ö, Vpp Å ±μ´¸É ´É ¤¨¶μ²Ó-¤¨¶μ²Ó´μ£μ ¢§ ¨³μ¤¥°¸É¢¨Ö.
‚ · ¡μÉ¥ [6] ¨§²μ¦¥´ ³¥Éμ¤ ¶μ²ÊÎ¥´¨Ö ±¨´¥É¨Î¥¸±μ£μ Ê· ¢´¥´¨Ö ¨§ Ê· ¢´¥´¨Ö ‹¨Ê¢¨²²Ö ¸ ¨¸Ìμ¤´Ò³ £ ³¨²ÓÉμ´¨ ´μ³ ¤²Ö ¸¨¸É¥³Ò ¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì Éμ³μ¢. ‚ ´ ¨¡μ²¥¥
μ¡Ð¥³ ¸²ÊÎ ¥ ±¨´¥É¨Î¥¸±μ¥ Ê· ¢´¥´¨¥ ³μ¦´μ § ¶¨¸ ÉÓ ¢ ¢¨¤¥
t
ρ̇ = i[HAA , ρ(t)] −
dt K̂(t )ρ(t − t ).
(2)
0
‡¤¥¸Ó ρ Å ³ É·¨Í ¶²μÉ´μ¸É¨ Éμ³μ¢, K̂ Å ´¥±¨° ¸Ê¶¥·μ¶¥· Éμ·, ±μÉμ·Ò° ´ §Ò¢ ¥É¸Ö
Ö¤·μ³ ¶ ³Öɨ.
·¥¤¶μ²μ¦¨³, ÎÉμ ¢·¥³Ö t t, É. ¥. ¢·¥³Ö ´ ¡²Õ¤¥´¨Ö § ¸¨¸É¥³μ° §´ Ψɥ²Ó´μ
¡μ²ÓÏ¥, Î¥³ Ì · ±É¥·´Ò° ¨´É¥·¢ ² ¶ ³Öɨ. ’죤 ³Ò ³μ¦¥³ · §²μ¦¨ÉÓ ¢ ·Ö¤ ³ É·¨ÍÊ
¶²μÉ´μ¸É¨ ¶μ¤ ¨´É¥£· ²μ³ ¨ μ£· ´¨Î¨ÉÓ¸Ö Éμ²Ó±μ ¤¢Ê³Ö β¥´ ³¨:
ρ(t − t ) ρ(t) −
∂ρ t.
∂t
(3)
¥·¢μ¥ ¸² £ ¥³μ¥ ¢ ÔÉμ³ · §²μ¦¥´¨¨ ¸μμÉ¢¥É¸É¢Ê¥É ³ ·±μ¢¸±μ³Ê ¶·¨¡²¨¦¥´¨Õ, Ê· ¢´¥´¨¥ ¤²Ö ±μÉμ·μ£μ Ìμ·μÏμ ¨§¢¥¸É´μ. ‚Éμ·μ¥ ¸² £ ¥³μ¥ ÊΨÉÒ¢ ¥É ¶μÖ¢²¥´¨¥ ±· É±μ¢·¥³¥´´μ° ¶ ³Öɨ.
μ¤¸É ¢²ÖÖ · §²μ¦¥´¨¥ (3) ¢ Ê· ¢´¥´¨¥ (2) ¨ ¸²¥¤ÊÖ ²μ£¨±¥ ¢Ò¢μ¤ ±¨´¥É¨Î¥¸±μ£μ
Ê· ¢´¥´¨Ö, ¶·¥¤²μ¦¥´´μ£μ ¢ ¸É ÉÓ¥ [6], ³μ¦´μ § ¶¨¸ ÉÓ ¥£μ ¢ ¸²¥¤ÊÕÐ¥³ ¢¨¤¥:
ρ̇ = i[HAA , ρ(t)] + L̂M ρ(t) + L̂N M
∂ρ
,
∂t
(4)
£¤¥ L̂M ¨ L̂N M Å ¸Ê¶¥·μ¶¥· Éμ·Ò, 춨¸Ò¢ ÕШ¥ ¸μμÉ¢¥É¸É¢¥´´μ ³ ·±μ¢¸±ÊÕ ¨ ´¥³ ·±μ¢¸±ÊÕ ·¥² ±¸ ͨÕ.
„ ´´μ¥ Ê· ¢´¥´¨¥ ´¥ ³μ¦¥É ¡ÒÉÓ ·¥Ï¥´μ, ¶μÔÉμ³Ê ¢ ¸É ÉÓ¥ [7] ¶·μ¨§¢μ¤´ÊÕ ¢ ¶· ¢μ° Î ¸É¨ Ê· ¢´¥´¨Ö ¶·¥¤²μ¦¨²¨ § ³¥´¨ÉÓ ³ ·±μ¢¸±¨³ β¥´μ³. Éμ μ¶· ¢¤ ´μ, ¥¸²¨
¥³ ·±μ¢¸± Ö ·¥² ±¸ ꬅ ±¢ ´Éμ¢ÒÌ ¸¨¸É¥³ 787
ÊÎ¥¸ÉÓ, ÎÉμ ¤²Ö ¡μ²ÓÏ¨Ì ¢·¥³¥´ ¤μ²¦´μ ¡ÒÉÓ ¸¶· ¢¥¤²¨¢μ ³ ·±μ¢¸±μ¥ Ê· ¢´¥´¨¥, ¢¸Ö
¨´Ëμ·³ Í¨Ö μ ±· É±μ¢·¥³¥´´μ° ¶ ³Öɨ ¸μ¤¥·¦¨É¸Ö ¢ ¸Ê¶¥·μ¶¥· Éμ·¥ L̂N M ¨ Éμ²Ó±μ ¢
´¥³, ¶μÔÉμ³Ê ¢ ¶· ¢μ° Î ¸É¨ Ê· ¢´¥´¨Ö ³μ¦´μ § ³¥´¨ÉÓ ¶·μ¨§¢μ¤´ÊÕ ³ ·±μ¢¸±¨³ β¥´μ³. μ¸²¥ § ³¥´Ò ³Ò ¶μ²ÊΨ³ ¸²¥¤ÊÕÐ¥¥ Ê· ¢´¥´¨¥, ±μÉμ·μ¥ ¤²Ö ´Ê²¥¢μ° É¥³¶¥· ÉÊ·Ò
·¥§¥·¢Ê · § ¶¨¸Ò¢ ¥É¸Ö ¢ ¢¨¤¥
ρ̇ = i
p=p
+
[σp+ σp− , ρ(t)]Vpp −
pp
γpp
(ρσp+ σp− + σp+ σp− ρ − σp− ρσp+ )+
2
i ∂γpp −2(σp+ σp− σi− ρσi+ − σi− ρσp+ σp+ σp− ) + [σi+ σi− σp+ σp− , ρ] , (5)
γii
4 ∂ω
ii pp
£¤¥
g12 = g21
3γ0
=
4
− [μ1 μ2 − (μ1 eR )(μ2 eR )]
cos (kR)
+
kR
+ [μ1 μ2 − 3(μ1 eR )(μ2 eR )]
sin (kR) cos (kR)
+
(kR)2
(kR)3
,
γ11 = γ22 = γ0 ,
γ12 = γ21 =
3γ0
2
sin (kR)
+
[μ1 μ2 − (μ1 eR )(μ2 eR )]
kR
+ [μ1 μ2 − 3(μ1 eR )(μ2 eR )]
cos (kR) sin (kR)
−
(kR)2
(kR)3
.
‡¤¥¸Ó γ0 Å ¸É ´¤ ·É´ Ö ±μ´¸É ´É ·¥² ±¸ ͨ¨, ±μÉμ· Ö ¶μ²ÊÎ ¥É¸Ö ¢ É¥μ·¨¨ μ¤´μ£μ
Éμ³ .
2. Š’“ ‹ˆˆˆ ˆ‡‹“—…ˆŸ
Š ± ¡Ò²μ ¶μ± § ´μ ´ ³¨ ¢ ¸É ÉÓ¥ [8], ±μ´ÉÊ· ²¨´¨¨ ¨§²ÊÎ¥´¨Ö μ¶·¥¤¥²Ö¥É¸Ö Ëμ·³Ê²μ°
⎧∞
⎨ 1
S(ω, Δk) = Re
σ1+ (t)σ1− (0) + σ2+ (t)σ2− (0) e−iωt dt +
⎩
π
0
⎫
∞
⎬
+
σ2 (t)σ1− (0) + σ1+ (t)σ2− (0) e−iωt dt ,
+ cos (ΔkR)
⎭
0
£¤¥ Δk = k − k Å · §´μ¸ÉÓ ¢μ²´μ¢ÒÌ ¢¥±Éμ·μ¢ ¶ ¤ ÕÐ¥° ¨ ¨§²ÊÎ¥´´μ° ¢μ²´; R Å
¢¥±Éμ·, ´ ¶· ¢²¥´´Ò° μÉ μ¤´μ£μ Éμ³ ± ¤·Ê£μ³Ê, ¥£μ ³μ¤Ê²Ó · ¢¥´ · ¸¸ÉμÖ´¨Õ ³¥¦¤Ê
Éμ³ ³¨.
ˆ¸¶μ²Ó§ÊÖ ±¢ ´Éμ¢ÊÕ É¥μ·¥³Ê ·¥£·¥¸¸¨¨ (¸³., ´ ¶·¨³¥·, [9]), ¨§ ·¥Ï¥´¨Ö Ê· ¢´¥´¨Ö (5) ³μ¦´μ ¶μ¸É·μ¨ÉÓ ±μ´ÉÊ· ²¨´¨¨ ¨§²ÊÎ¥´¨Ö ¢ ´ ²¨É¨Î¥¸±μ³ ¢¨¤¥, μ¤´ ±μ ¶μ²ÊÎ¥´´Ò¥ Ëμ·³Ê²Ò ¸²¨Ï±μ³ £·μ³μ§¤±¨ ¨ §¤¥¸Ó ´¥ ¶·¨¢μ¤ÖɸÖ. ‘μμÉ¢¥É¸É¢ÊÕШ¥ £· ˨±¨
¶·¥¤¸É ¢²¥´Ò ´ ·¨¸Ê´±¥.
788 ‘¥³¨´ ‚. ‚., ƒμ·μÌμ¢ . ‚.
Šμ´ÉÊ· ²¨´¨¨ ¨§²ÊÎ¥´¨Ö ¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì ¤¢ÊÌÊ·μ¢´¥¢ÒÌ Éμ³μ¢ ¢ ´¥³ ·±μ¢¸±μ³ ¸²ÊÎ ¥. · ³¥É·Ò ¢ ¸¨¸É¥³¥ kR = π/5, ΔkR = π/6. ‘¶²μÏ´ Ö ±·¨¢ Ö ¸μμÉ¢¥É¸É¢Ê¥É γ0 (∂γ0 /∂ω) = 0
(³ ·±μ¢¸±¨° ¸²ÊÎ °), ÏÉ·¨Ìμ¢ Ö Å γ0 (∂γ0 /∂ω) = 2, ÏÉ·¨Ì¶Ê´±É¨·´ Ö Å γ0 (∂γ0 /∂ω) = 5
•μ·μÏμ ¢¨¤´μ, ÎÉμ ¸ ·μ¸Éμ³ ¶ · ³¥É· ´¥³ ·±μ¢μ¸É¨ γ0 (∂γ0 /∂ω) ±μ´ÉÊ· ¤¥Ëμ·³¨·Ê¥É¸Ö ¨ ¥£μ ³ ±¸¨³Ê³Ò ¸³¥Ð ÕɸÖ. Éμ, ¢¨¤¨³μ, ¸¢Ö§ ´μ ¸ ¤¥Ëμ·³ ͨ¥° ¸É·Ê±ÉÊ·Ò
Ô´¥·£¥É¨Î¥¸±¨Ì Ê·μ¢´¥°, ± ±μÉμ·μ° ¶·¨¢μ¤ÖÉ ÔËË¥±ÉÒ ¶ ³Öɨ.
‡Š‹—…ˆ…
‚ · ¡μÉ¥ · ¸¸³μÉ·¥´ ³μ¤¥²Ó ¤¢ÊÌ ¨¤¥´É¨Î´ÒÌ ¤¨¶μ²Ó-¤¨¶μ²Ó´μ-¢§ ¨³μ¤¥°¸É¢ÊÕШÌ
Éμ³μ¢. ‚ · ³± Ì ±¢ ´Éμ¢μ° É¥μ·¨¨ μɱ·ÒÉÒÌ ¸¨¸É¥³ ¢Ò¢¥¤¥´μ 춥· Éμ·´μ¥ ±¨´¥É¨Î¥¸±μ¥ Ê· ¢´¥´¨¥, ¢ ±μÉμ·μ³ ÊÎÉ¥´Ò ÔËË¥±ÉÒ ±· É±μ¢·¥³¥´´μ° ¶ ³Öɨ. μ¸´μ¢¥ ·¥Ï¥´¨Ö
¶μ²ÊÎ¥´´μ£μ Ê· ¢´¥´¨Ö ¶μ¸É·μ¥´ ±μ´ÉÊ· ²¨´¨¨ ¨§²ÊÎ¥´¨Ö. μ± § ´μ, ÎÉμ ÔËË¥±ÉÒ ¶ ³Öɨ ¶·¨¢μ¤ÖÉ ± ¸ÊÐ¥¸É¢¥´´μ° ¤¥Ëμ·³ ͨ¨ ±μ´ÉÊ· ¨§²ÊÎ¥´¨Ö ¢ ¸· ¢´¥´¨¨ ¸ ³ ·±μ¢¸±¨³
¸²ÊÎ ¥³.
‘ˆ‘Š ‹ˆ’…’“›
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2002.
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V. 48, No. 2. P. 119Ä130.
¥³ ·±μ¢¸± Ö ·¥² ±¸ ꬅ ±¢ ´Éμ¢ÒÌ ¸¨¸É¥³ 789
6. Kurizki G., Ben-Reuven A. Theory of Cooperative Fluorescence from Products of Reactions or
Collisions: Identical Neutral Atomic Fragments // Phys. Rev. A. 1987. V. 36. P. 90Ä102.
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Rev. A. 1992. V. 46, No. 3. P. 1507Ä1515.
8. ƒμ·μÌμ¢ . ‚., ‘¥³¨´ ‚. ‚. ¸Î¥É ¸¶¥±É· ˲Êμ·¥¸Í¥´Í¨¨ ¤²Ö ¤¢ÊÌ ¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì Éμ³μ¢ // ¶É¨± ¨ ¸¶¥±É·μ¸±μ¶¨Ö. 2009. ’. 107, º 4. ‘. 617Ä622.
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